How to remove a subdivision of a cube
Select the seven verts of the corner. Select the lower right vertex last so it becomes the active (white) vertex.



Extrude the selection with EESCAPE. That way the newly created verts stay in their places. Set the pivot point to 'Active Object' and scale the selection to 0 on the Y axis SY0RETURN. This will set all vertices to the y value of the active vertex, putting them in the back. Depending on how your object is oriented, you might need to pick a different axis.



Select everything with AA, remove doubles with WR and recalculate normals with CTRL-N